Stabilized reduced basis approximation of incompressible three-dimensional Navier–Stokes equations in parametrized deformed domains

Simone Deparis
Springer Verlag, 2012
Article

Résumé

In this work we are interested in the numerical solution of the steady incompressible Navier-Stokes equations for fluid flow in pipes with varying curvatures and cross-sections. We intend to compute a reduced basis approximation of the solution, employing the geometry as a parameter in the reduced basis method. This has previously been done in a spectral element $P_{N} - P_{N-2}$ setting in two dimensions for the steady Stokes equations. To compute the necessary basis-functions in the reduced basis method, we propose to use a stabilized $P_1 - P_1$ finite element method for solving the Navier-Stokes equations on different geometries. By employing the same stabilization in the reduced basis approximation, we avoid having to enrich the velocity basis in order to satisfy the inf-sup condition. This reduces the complexity of the reduced basis method for the Navier-Stokes problem, while keeping its good approximation properties. We prove the well posedness of the reduced problem and present numerical results for selected parameter dependent three dimensional pipes.

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https://infoscience.epfl.ch/record/148584?ln=fr

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Simone Deparis
Springer Verlag, 2012
Article

Résumé

Official source

https://infoscience.epfl.ch/record/148584?ln=fr

À propos de ce résultat

Concepts associés (16)

Concepts associés

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In the last two decades, the level set method has been extensively used for the numerical solution of interface problems in different domains. The basic idea is to embed the interface as the level set of a regular function. In this paper we focus on the numerical solution of interface advection equations appearing in free-surface fluid dynamics problems, where naive finite element implementations are unsatisfactory. As a matter of fact, practitioners in fluid dynamics often complain that the mass of each fluid component is not conserved, a phenomenon which is therefore often referred to as mass loss. In this paper we propose and compare two finite element implementations that cure this ill-behaviour without the need to resort to combined strategies (such as e.g. particle level set). The first relies on a discontinuous Galerkin discretization, which is known to give very good performance when facing hyperbolic problems; the second is a stabilized continuous FEM implementation based on the stabilization method presented in [N. Parolini, Computational fluid dynamics for Naval engineering applications, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, June 2004], which is free from many of the problems that classical methods exhibit when applied to unsteady problems. [All rights reserved Elsevier]

2006

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Isogeometric analysis (IGA) is a computational methodology recently developed to numerically approximate Partial Differential Equation (PDEs). It is based on the isogeometric paradigm, for which the same basis functions used to represent the geometry are then used to approximate the unknown solution of the PDEs. In the case in which Non-Uniform Rational B-Splines (NURBS) are used as basis functions, their mathematical properties lead to appreciable benefits for the numerical approximation of PDEs, especially for high order PDEs in the standard Galerkin formulation. In this framework, we propose an a priori error estimate, extending existing results limited to second order PDEs. The improvements in both accuracy and efficiency of IGA compared to Finite Element Analysis (FEA), encourage the use of this methodology in the haemodynamic applications. In fact, the simulation of blood flow in arteries requires the numerical approximation of Fluid-Structure Interaction (FSI) problems. In order to account for the deformability of the vessel, the Navier-Stokes equations representing the blood flows, are coupled with structural models describing the mechanical response of the arterial wall. However, the FSI models are complex from both the mathematical and the numerical points of view, leading to high computational costs during the simulations. With the aim of reducing the complexity of the problem and the computational costs of the simulations, reduced FSI models can be considered. A first simplification, based on the assumption of a thin arterial wall structure, consists in considering shell models to describe the mechanical properties of the arterial walls. Moreover, by means of the additional kinematic condition (continuity of velocities) and dynamic condition (balance of contact forces), the structural problem can be rewritten as generalized boundary condition for the fluid problem. This results in a generalized Navier-Stokes problem which can be expressed only in terms of the primitive variables of the fluid equations (velocity and pressure) and in a fixed computational domain. As a consequence, the computational costs of the numerical simulations are significantly reduced. On the other side, the generalized boundary conditions associated to the reduced FSI model could involve high order derivatives, which need to be suitably approximated. With this respect, IGA allows an accurate, straightforward and efficient numerical approximation of the generalized Navier-Stokes equations characterizing the reduced FSI problem. In this work we consider the numerical approximation of reduced FSI models by means of IGA, for which we discuss the numerical results obtained in Haemodynamic applications.

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A posteriori error estimation for partial differential equations with random input data

Diane Sylvie Guignard

This thesis is devoted to the derivation of error estimates for partial differential equations with random input data, with a focus on a posteriori error estimates which are the basis for adaptive strategies. Such procedures aim at obtaining an approximation of the solution with a given precision while minimizing the computational costs. If several sources of error come into play, it is then necessary to balance them to avoid unnecessary work. We are first interested in problems that contain small uncertainties approximated by finite elements. The use of perturbation techniques is appropriate in this setting since only few terms in the power series expansion of the exact random solution with respect to a parameter characterizing the amount of randomness in the problem are required to obtain an accurate approximation. The goal is then to perform an error analysis for the finite element approximation of the expansion up to a certain order. First, an elliptic model problem with random diffusion coefficient with affine dependence on a vector of independent random variables is studied. We give both a priori and a posteriori error estimates for the first term in the expansion for various norms of the error. The results are then extended to higher order approximations and to other sources of uncertainty, such as boundary conditions or forcing term. Next, the analysis of nonlinear problems in random domains is proposed, considering the one-dimensional viscous Burgers' equation and the more involved incompressible steady-state Navier-Stokes equations. The domain mapping method is used to transform the equations in random domains into equations in a fixed reference domain with random coefficients. We give conditions on the mapping and the input data under which we can prove the well-posedness of the problems and give a posteriori error estimates for the finite element approximation of the first term in the expansion. Finally, we consider the heat equation with random Robin boundary conditions. For this parabolic problem, the time discretization brings an additional source of error that is accounted for in the error analysis. The second part of this work consists in the analysis of a random elliptic diffusion problem that is approximated in the physical space by the finite element method and in the stochastic space by the stochastic collocation method on a sparse grid. Considering a random diffusion coefficient with affine dependence on a vector of independent random variables, we derive a residual-based a posteriori error estimate that controls the two sources of error. The stochastic error estimator is then used to drive an adaptive sparse grid algorithm which aims at alleviating the so-called curse of dimensionality inherent to tensor grids. Several numerical examples are given to illustrate the performance of the adaptive procedure.

EPFL2016

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A posteriori error estimation for partial differential equations with random input data

Diane Sylvie Guignard

EPFL2016

2012